Measurements of jet activity in top-quark pair events produced in proton–proton collisions are presented, using 3.2 fb−1 of pp collision data at a centre-of-mass energy of 13 TeV collected by the ATLAS experiment at the Large Hadron Collider. Events are chosen by requiring an opposite-charge eμ pair and two b-tagged jets in the final state. The normalised differential cross-sections of top-quark pair production are presented as functions of additional-jet multiplicity and transverse momentum, pT. The fraction of signal events that do not contain additional jet activity in a given rapidity region, the gap fraction, is measured as a function of the pT threshold for additional jets, and is also presented for different invariant mass regions of the eμbb¯ system. All measurements are corrected for detector effects and presented as particle-level distributions compared to predictions with different theoretical approaches for QCD radiation. While the kinematics of the jets from top-quark decays are described well, the generators show differing levels of agreement with the measurements of observables that depend on the production of additional jets.